In this section, through figuring out the minimax lower bounds of the estimation error, we argue that it is essential to incorporate sparsity information of \((\theta _0,\Sigma )\)  in \((\widehat{\theta }_0,\widehat{\Sigma })\)  in sparse model. For minimax lower bound in standard learning problems, studies can be found in
[1]}, [2]} for dense case and [3]}, [4]}, [5]} for sparse case.
